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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfon2lem1 | Structured version Visualization version GIF version |
Description: Lemma for dfon2 35774. (Contributed by Scott Fenton, 28-Feb-2011.) |
Ref | Expression |
---|---|
dfon2lem1 | ⊢ Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | truni 5281 | . 2 ⊢ (∀𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)}Tr 𝑦 → Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)}) | |
2 | nfsbc1v 3811 | . . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
3 | nfv 1912 | . . . . 5 ⊢ Ⅎ𝑥Tr 𝑦 | |
4 | nfsbc1v 3811 | . . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜓 | |
5 | 2, 3, 4 | nf3an 1899 | . . . 4 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦 ∧ [𝑦 / 𝑥]𝜓) |
6 | vex 3482 | . . . 4 ⊢ 𝑦 ∈ V | |
7 | sbceq1a 3802 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
8 | treq 5273 | . . . . 5 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦)) | |
9 | sbceq1a 3802 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ [𝑦 / 𝑥]𝜓)) | |
10 | 7, 8, 9 | 3anbi123d 1435 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ Tr 𝑥 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦 ∧ [𝑦 / 𝑥]𝜓))) |
11 | 5, 6, 10 | elabf 3676 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} ↔ ([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦 ∧ [𝑦 / 𝑥]𝜓)) |
12 | 11 | simp2bi 1145 | . 2 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} → Tr 𝑦) |
13 | 1, 12 | mprg 3065 | 1 ⊢ Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1086 ∈ wcel 2106 {cab 2712 [wsbc 3791 ∪ cuni 4912 Tr wtr 5265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-v 3480 df-sbc 3792 df-ss 3980 df-uni 4913 df-iun 4998 df-tr 5266 |
This theorem is referenced by: dfon2lem3 35767 dfon2lem7 35771 |
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